From [1]}), we can find \(T_1>0\)  satisfying that
\({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x\varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\le C(\rho ,T_1).\) 
Therefore, it remains to guarantee that
\(\mathcal {A}+ \mathcal {B}+ \mathcal {C}:={\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert  \partial _x \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert } +{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x^2\varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+ \Vert |x|^{r}\varphi \Vert _{L^\infty _T L^2_x}<C(\rho ,T),\) 
for some \(t \in (0,T_1]\)  sufficiently small.
Since, \(\psi \)  solves the integral equation (REF ) (recall the remark at the begining of the proof of Proposition REF ), we have that
\(\mathcal {A}&\lesssim \Vert  e^{it \partial _x^2}(\partial _x\psi _0)\Vert _{L_T^{\infty }L^2_x}+\Vert  e^{it \partial _x^2}(\partial _x \psi _0)\Vert _{L_T^{4}L^\infty _x} \nonumber \\& \qquad +\Big \Vert \int _{0}^te^{i(t-t^{\prime }) \partial _x^2}\partial _x (\psi ^2\overline{\varphi })(x,t^{\prime })\, dt^{\prime } \Big \Vert _{L_T^{\infty }L^2_x} \nonumber \\& \qquad +\Big \Vert \int _{0}^te^{i(t-t^{\prime }) \partial _x^2}\partial _x(\psi ^2\overline{\varphi })(x,t^{\prime })\, dt^{\prime } \Big \Vert _{L_T^{4}L^\infty _x} \nonumber \\& =:\mathcal {A}_1+ \mathcal {A}_2+ \mathcal {A}_3+ \mathcal {A}_4.\) 

From, (REF ) we immediately obtain
\(\mathcal {A}_1 + \mathcal {A}_2\lesssim \Vert \partial _x\psi _0\Vert _{L^2}\lesssim \Vert \psi _0\Vert _{H^1} < \rho .\) 
Similarly, using
(REF )
and
(REF ) we get
\(\mathcal {A}_3 + \mathcal {A}_4&\lesssim \Vert \partial _x(\psi ^2\overline{\varphi })\Vert _{L^1_T L^2_x}\lesssim T\Vert \psi ^2\overline{\varphi }\Vert _{L^\infty _T H^1_x}\lesssim T\Vert \varphi \Vert _{L^\infty _T H^1_x}\Vert \psi \Vert _{L^\infty _T H^1_x}^2 \nonumber \\& \sim T\Big (\Vert \varphi \Vert _{L^\infty _T L^2_x}+ \Vert \partial _x \varphi \Vert _{L^\infty _T L^2_x}\Big )\Big (\Vert \psi \Vert _{L^\infty _T L^2_x}^2+ \Vert \partial _x \psi \Vert _{L^\infty _T L^2_x}^2\Big ) \nonumber \\& \lesssim T\Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\Big )\Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }^2+ {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }^2\Big ).\) 

Combining
(REF ), (REF ) and (REF )
we deduce, for any \(T<T_1\) ,
\(\mathcal {A}\lesssim \rho +T \,C(\rho ,T_1)\Big (C(\rho ,T_1)^2+ \mathcal {A}^2\Big ).\) 
Hence, it is possible to choose \(T_2<T_1\)  sufficiently small verifying that
\(\mathcal {A}\lesssim C(\rho ,T_2).\) 
   
For the analysis of the factor \(\mathcal {B}\) , we apply (REF ) to write
\(\mathcal {B}\lesssim {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x\psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x(|\varphi |^2 \varphi )\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }=: \mathcal {B}_1 + \mathcal {B}_2.\) 
Notice that \(\mathcal {B}_1=\mathcal {A}\) , treated above. On the other hand, (REF ) yields to
\(\mathcal {B}_2& \lesssim \Vert \varphi ^2\partial _x \overline{\varphi }\Vert _{L^{\infty }_T L^2_x}+\Vert \varphi ^2\partial _x \overline{\varphi }\Vert _{L^{4}_TL^{\infty }_x}+\Vert |\varphi |^2\partial _x \varphi \Vert _{L^{\infty }_T L^2_x}+\Vert  |\varphi |^2\partial _x \varphi \Vert _{L^{4}_TL^{\infty }_x} \nonumber \\& \lesssim \Vert \partial _x \varphi \Vert _{L^{\infty }_T L^2_x}\Vert \varphi \Vert ^2_{L^{\infty }_T H^1_x}+\Vert \partial _x \varphi \Vert _{L^{4}_T L^\infty _x}\Vert \varphi \Vert ^2_{L^{\infty }_T H^1_x} \nonumber \\& \lesssim {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }^2+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x\varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }^2\Big ).\) 

Thus, (REF ) and (REF ) imply
\(\mathcal {B}\lesssim C(\rho ,T_2).\) 
   
We finally consider the term
\(\mathcal {C}\) . Using this time the integral equation (REF ), one realizes that
\(\mathcal {C}& \lesssim \Vert |x|^{r} e^{it \partial _x^2}\varphi _0 \Vert _{L^\infty _T L^2_x}+\Big \Vert |x|^{r} \int _{0}^te^{i(t-t^{\prime }) \partial _x^2}(\varphi ^2 \overline{\psi })(x,t^{\prime })\, dt^{\prime } \Big \Vert _{L^\infty _T L^2_x}=: \mathcal {C}_1+ \mathcal {C}_2.\) 

Next, similarly to (REF ) we deduce, for any \(T<T_2\) ,
\(\mathcal {C}_1&\lesssim (T^{r/2}+T^m)\Vert \varphi _0\Vert _{H^1}+\Vert |x|^{r}\varphi _0\Vert _{L^2}\lesssim C(\rho , T_2).\) 

As for \(\mathcal {C}_2\)  we proceed as in (REF ) above to get
\(\mathcal {C}_2&\lesssim T \,\Vert \varphi \Vert _{L^\infty _T H_x^{1}} \,\Vert  \psi \Vert _{L^\infty _T H_x^{1}} \,\Vert |x|^{r}\varphi \Vert _{L^\infty _T L_x^2}+(T^{1+r/2} + T^{1+r}) \,\,\Vert \varphi \Vert _{L^\infty _TH^1_{x}}^2\Vert \psi \Vert _{L^\infty _TH^1_{x}}\nonumber \\& \lesssim T \,\Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\Big )\Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\Big )\Vert |x|^{r}\varphi \Vert _{L^\infty _T L_x^2} \nonumber \\& \qquad +(T^{1+r/2} + T^{1+r}) \,\Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }^2+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \varphi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }^2\Big )\Big ({\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }+{\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert \partial _x \psi \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }\Big ).\) 

Therefore, putting together (REF ), (REF ),
(REF ) and (REF ), one sees that for \(T<T_2\)
\(\mathcal {C}\lesssim C(\rho ,T_2)+ T C(\rho ,T_2) \mathcal {C}.\) 

Thus, taking \(T_3<T_2\)  small enough, we conclude
\(\mathcal {C}\lesssim C(\rho ,T_3).\) 
   
In summary, our goal (REF ) is a consequence of (REF ), (REF ) and (REF ).
